In this talk we will first discuss the Dyson-Taylor Commutator method for short time asymptotics that we did not finish last time. And then we will turn to some practical issues, namely, local volatilities and implied volatilities. Recall that in a local volatility model, the volatility term \sigma(t,S) is assumed a function of time and the underlying asset price. However, \sigma(t,S) is not observable in the market and only option prices are observable. So looking for local volatilities is an inverse problem, i.e., we need to deduce \sigma(t,S) from market option prices. The famous Dupire's work provides us with analytic formulas, though it has many disadvantages.
Implied volatilities are also intensively used by traders. They usually form a surface called volatility smile. In
general, we have no closed-form formulas for the smile. But the short time limit is in analytic form by the work of Berestycki, Busca and Florent. And this is the starting point for many asymptotic implied volatilities, including Dan Parjol's talk in a few weeks.